There are certain rules you are expected to follow when your ball is on or near the green.
Suppose one golfer in a Par-3 hole hits his ball into a sandtrap not far from the pin. Another player lands his ball on the green but it is several feet farther away from the pin than the first ball. Who shoots first? According to the U.S. Golf Association, the ball that's farthest from the hole is always played first. So in this case the ball on the green would be played first.
What about finishing out the hole? If the player in the trap gets lucky and places his ball near the cup, with only a tap-in left, can he finish it off before the other player tries his putt? It depends on whether they're involved in stroke play or match play. In either competition there is no penalty for playing out of turn. However, in match play his opponent has the option to recall the stroke and make the player finish his shot in order.